1.3 Number System Conversions

Understanding how to represent numbers is fundamental to digital systems, and various number systems exist, each with a unique base or radix. While humans commonly use the decimal (base-10) system, computers primarily operate using binary (base-2). To bridge this gap and facilitate efficient data processing and communication, it's essential to master the art of number system conversions. This section will delve into the systematic methods for converting numbers between different bases, including binary, octal, decimal, and hexadecimal, providing the tools necessary to translate values across these crucial representations.

Part 1.3.1 Conversion from Decimal to Binary Numbers

There are two methods for converting DEC to BIN: The Direct Method and Successive Division.

Part 1.3.1.a Direct Method

This is the step by step procedure using a direct method.

Step 1: Repeatedly subtract the largest power of 2 that is less than or equal to the decimal number.

Step 2: Continue this process until the remainder is zero. Note: A negative remainder is not allowed.

Step 3: For each power of 2 subtracted, write a '1'.

Step 4: List the '1's and '0's in the order in which the subtractions were performed, starting with the highest power of 2 and proceeding to the lowest."

Example No. 1

Convert 55 to BIN:

Hint: The largest power of 2 we can use is 2⁵ = 32.

Minuend	Subtrahend	Difference	Binary Equivalent
55	32	23	1
23	16	7	1
7	8	-1	0
7	4	3	1
3	2	1	1
1	1	0	1

Read it from top to bottom, the answer is 110111₂

Example No. 2

Convert 78 to BIN:

Hint: The largest power of 2 we can use is 2⁶ = 64.

Minuend	Subtrahend	Difference	Binary Equivalent
78	64	14	1
14	32	-18	0
14	16	-2	0
14	8	6	1
6	4	2	1
2	2	0	1
0	1	-1	0

Read it from top to bottom, the answer is 100110₂

Example No. 3

Convert 345 to BIN:

Hint: The largest power of 2 we can use is 2⁸ = 256.

Minuend	Subtrahend	Difference	Binary Equivalent
345	256	89	1
89	128	-39	0
89	64	25	1
25	32	-7	0
25	16	9	1
9	8	1	1
1	4	-3	0
1	2	-1	0
1	1	0	1

Read it from top to bottom, the answer is 1010110011₂

Part 1.3.1.b Successive Division Method

 This is the step by step procedure using a Successive Division method.

Step 1: Successively divide the decimal number by 2.

Step 2: Place the resulting quotient directly below the dividend.

Step 3: Place the remainder opposite the quotient for clarity.

Step 4: Repeat steps 1-3 until the quotient is less than 2.

Step 5: The equivalent binary number is formed by reading the remainders from bottom to top. The first remainder is the least significant bit (LSB), while the last remainder is the most significant bit (MSB).

Example No. 4

Convert 55 to BIN:

Dividend	Divisor	Quotient	Remainder
55	2	27	1
27	2	13	1
13	2	6	1
6	2	3	0
3	2	1	1
1	2	0	1

Read it from bottom to top, the answer is 110111_2.

Example No. 5

Convert 78 to BIN:

Dividend	Divisor	Quotient	Remainder
78	2	39	0
39	2	19	1
19	2	9	1
9	2	4	1
4	2	2	0
2	2	1	0
1	2	0	1

Read it from bottom to top, the answer is 1001110_2.

Example No. 6

Convert 345 to BIN:

Dividend	Divisor	Quotient	Remainder
345	2	172	1
172	2	86	0
86	2	43	0
43	2	21	1
21	2	10	1
10	2	5	0
5	2	2	1
2	2	1	0
1	2	0	1

Read it from bottom to top, the answer is 101011001₂

Part 1.3.2 Conversion from Binary to Octal Numbers

To convert a binary number to octal, we must group the binary digits into groups of three, starting from the right. Then, convert each group into its equivalent decimal number.

Example No. 6

Convert 100011010101010₂ to OCT

Solution:

Group the number into group of three

100 011 010 101 010

	5^th 3 bit	4^th 3 bit	3^rd 3 bit	2^nd 3 bit	1^st 3 bit
	100	011	010	101	010
3 bit binary string to Decimal	4	3	2	5	2

Read it from left to right, the answer is: 43252₈

Example No. 7

Convert 11101010101111010101₂ to OCT

Solution:

Group the number into group of three

011 101 010 101 111 010 101

	7^th 3bit	6^th 3bit	5^th 3bit	4^th 3bit	3^rd 3bit	2^nd 3bit	1^st 3bit
	011	101	010	101	111	010	101
3 bit binary string to Decimal	3	5	2	5	7	2	5

Read it from left to right, the answer is: 3525725₈

Example No. 8

Convert 11111111111111111111111₂ to OCT

Solution:

Group the number into group of three

011 111 111 111 111 111 111 111

	8^th 3bit	7^th 3bit	6^th 3bit	5^th 3bit	4^th 3bit	3^rd 3bit	2^nd 3bit	1^st 3bit
	011	111	111	111	111	111	111	111
3 bit binary string to Decimal	3	7	7	7	7	7	7	7

Read it from left to right, the answer is: 3777777₈

Part 1.3.3 Conversion from Octal to Binary Numbers

Converting octal to binary is the reverse process of converting binary to octal. To convert octal to binary, first convert each octal digit to its equivalent 3-bit binary representation, then concatenate the resulting binary strings.

Example No. 9

Convert 43252₈ to BIN

	5^th digit	4^th digit	3^rd digit	2^nd digit	1^st digit
	4	3	2	5	2
Decimal to 3bit Binary String	100	011	010	101	010

Read it from left to right, the answer is: 100011010101010₂

Example No. 10

Convert 3525725₈ to BIN

	7^th digit	6^th digit	5^th digit	4^th digit	3^rd digit	2^nd digit	1^st digit
	3	5	2	5	7	2	5
Decimal to 3bit Binary String	011	101	010	101	111	010	101

Read it from left to right, the answer is: 11101010101111010101₂

Example No. 11

Convert 3777777₈ to BIN

	7^th digit	6^th digit	5^th digit	4^th digit	3^rd digit	2^nd digit	1^st digit
	3	7	7	7	7	7	7
Decimal to 3bit Binary String	011	111	111	111	111	111	111

Read it from left to right, the answer is: 11111111111111111111111₂

Part 1.3.4 Conversion from Binary to Hexadecimal Numbers

To convert a binary number to octal, we must group the binary digits into groups of four, starting from the right. Then, convert each group into its equivalent decimal number.

Example No. 12

Convert 1110101000101010101010001₂ to Hexadecimal

Group the number into group of three

0001 1101 0100 0101 0101 0101 0001

	7^th 4 string bit	6^th 4 string bit	5^th 4 string bit	4^th 4 string bit	3^rd 4 string bit	2^nd 4 string bit	1^st 4 string bit
	0001	1101	0100	0101	0101	0101	0001
4 bit binary string to Decimal	1	D	4	5	5	5	1

Read it from left to right, the answer is: 1D45551₁₆

Example No. 13

Convert 1110111100011100011111000111100011111₂ to Hexadecimal

Group the number into group of three

1111 0001 1111 1000 1111 1000 0011 1110 1101 0001

	7^th 4 string bit	7^th 4 string bit	7^th 4 string bit	7^th 4 string bit	6^th 4 string bit	5^th 4 string bit	4^th 4 string bit	3^rd 4 string bit	2^nd 4 string bit	1^st 4 string bit
	1111	0001	1111	1000	1111	1000	0011	1110	1101	0001
4 bit binary string to Decimal	F	1	F	8	F	8	3	E	D	1

Read it from left to right, the answer is: 1DE38F8F1F₁₆

Part 1.3.5 Conversion from Hexadecimal to Binary Numbers

Converting hex to binary is the reverse process of converting binary to hex. To convert hex to binary, first convert each hex digit to its equivalent 4-bit binary representation, then concatenate the resulting binary strings.

Example No. 14

Convert 1D45551₁₆ to Binary

	7^th digit	6^th digit	5^th digit	4^th digit	3^rd digit	2^nd digit	1^st digit
	1	D	4	5	5	5	1
Decimal to 4bit Binary String	0001	1101	0100	0101	0101	0101	0001

Read it from left to right, the answer is: 0001110101000101010101010001₂

Example No. 15

Convert 1628DDF₁₆ to Binary

	7^th digit	6^th digit	5^th digit	4^th digit	3^rd digit	2^nd digit	1^st digit
	1	6	2	8	D	D	F
Decimal to 4bit Binary String	0001	0110	0010	1000	1101	1101	1111

Read it from left to right, the answer is: 0001011000101000110111011111₂

Part 1.3.6 Conversion from Decimal to Octal Numbers

Step 1: Successively divide the decimal number by 8.

Step 2: Place the resulting quotient directly below the dividend.

Step 3: Place the remainder opposite the quotient for clarity.

Step 4: Repeat steps 1-3 until the quotient is less than 2.

Step 5: The equivalent octal number is formed by reading the remainders from bottom to top. The first remainder is the least significant bit (LSB), while the last remainder is the most significant bit (MSB).

Example No. 16

Convert 456 to Octal number:

Dividend	Divisor	Quotient	Remainder
456	8	57	0
57	8	7	1
7	8	0	7

Let’s read it from bottom to top, the answer is: 710₈

Example No. 17

Convert 1453 to Octal number:

Dividend	Divisor	Quotient	Remainder
1453	8	181	5
181	8	22	5
22	8	2	6
2	8	0	2

Let’s read it from bottom to top, the answer is: 2655₈

Example No. 18

Convert 8595 to Octal number:

Dividend	Divisor	Quotient	Remainder
8595	8	1074	3
1074	8	134	2
134	8	16	6
16	8	2	0
2	8	0	2

Let’s read it from bottom to top, the answer is: 20623₈

Part 1.3.7 Conversion from Decimal to Hexadecimal Numbers

Step 1: Successively divide the decimal number by 16.

Step 2: Place the resulting quotient directly below the dividend.

Step 3: Place the remainder opposite the quotient for clarity.

Step 4: Repeat steps 1-3 until the quotient is less than 2.

Step 5: The equivalent oct number is formed by reading the remainders from bottom to top. The first remainder is the least significant bit (LSB), while the last remainder is the most significant bit (MSB).

Example No. 19

Convert 456 to Hexadecimal number:

Dividend	Divisor	Quotient	Remainder
456	16	28	8
28	16	1	C
1	16	0	1

Let’s read it from bottom to top, the answer is: 1C8₁₆

Example No. 20

Convert 75839 to Hexadecimal number:

Dividend	Divisor	Quotient	Remainder
75839	16	4739	F
4739	16	296	3
296	16	18	8
18	16	1	2
1	16	0	1

Let’s read it from bottom to top, the answer is: 1283F₁₆